What is the derivative of $$y = \text{cos}^5(x)$$ ?

$$\frac{dy}{dx} = -5\text{cos}^4(x)\text{sin}(x)$$ .

How do you differentiate $$y = \text{cos}^5(4x^3)$$ using the Chain Rule?

$$\frac{dy}{dx} = -60x^2\text{cos}^4(4x^3)\text{sin}(4x^3)$$ .

What is the derivative of $$y = \text{tan}(3x^2 + 2)$$ ?

$$\frac{dy}{dx} = 6x\text{sec}^2(3x^2 + 2)$$ .

Find the derivative of $$y = \text{sin}(x^3 - 2x)$$ .

$$\frac{dy}{dx} = (3x^2 - 2)\text{cos}(x^3 - 2x)$$ .

What is the derivative of $$y = (csc(x) + 1)^7$$ ?

$$\frac{dy}{dx} = -7(csc(x) + 1)^6(csc(x)cot(x))$$ .

What is the general form of the Chain Rule?

If $$y = f(g(x))$$ , then $$\frac{dy}{dx} = f$$ .

Provide an example of a function that requires the Chain Rule to differentiate.

An example is $$y = \text{sin}(3x^2 + 1)$$ . The derivative is $$\frac{dy}{dx} = \text{cos}(3x^2 + 1) \times 6x$$ .

What is the derivative of $$y = e^{g(x)}$$ ?

$$\frac{dy}{dx} = e^{g(x)} \times g$$ .

How do you apply the Chain Rule to $$y = \text{ln}(g(x))$$ ?

$$\frac{dy}{dx} = \frac{1}{g(x)} \times g$$ .

What is the derivative of $$y = \text{cos}(h(x))$$ ?

$$\frac{dy}{dx} = -\text{sin}(h(x)) \times h$$ .

Explain how to differentiate a function of the form $$y = f(g(h(x)))$$ .

Use the Chain Rule multiple times: $$\frac{dy}{dx} = f$$ .

What is the significance of the derivative in real-world applications?

Derivatives represent rates of change, which are crucial in fields such as physics, engineering, and economics for understanding how quantities change in relation to one another.

DERIVATIVE CHAIN RULE Flashcards

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1.

FLASHCARD QUESTION

Front

Define the Chain Rule in calculus.

Back

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It states that if a function y = f(g(x)), then the derivative dy/dx is given by dy/dx = f'(g(x)) * g'(x).

2.

FLASHCARD QUESTION

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3.

FLASHCARD QUESTION

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4.

FLASHCARD QUESTION

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5.

FLASHCARD QUESTION

Front

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6.

FLASHCARD QUESTION

Front

Back

7.

FLASHCARD QUESTION

Front

Explain the importance of the Chain Rule in calculus.

Back

The Chain Rule is essential for differentiating composite functions, which are common in calculus. It allows for the calculation of derivatives in a systematic way, facilitating the analysis of complex functions.

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